\(\[\begin{align} & \int\limits_{C}{{\vec{F}}}\cdot d\vec{r} \\ & \int\limits_{C}{(x+yz)}dx+2xdy+xyzdz \\ & \vec{F}(x,y,z)=P(x,y,z)\vec{i}+Q(x,y,z)\vec{j}+R(x,y,z)\vec{k}\vec{F}(x,y,z) \\ & (x+yz)\vec{i}+2x\vec{j}+xyz\vec{k} \\ & \left\{ \begin{matrix} P(x,y,z) & = & x+yz \\ {} & {} & {} \\ Q(x,y,z) & = & 2x \\ {} & {} & {} \\ R(x,y,z) & = & xyz \\ \end{matrix} \right. \\ & \\ & \\ \end{align}\] \)

Expandindo o produto escalar:

\(\[\begin{align} & =\int\limits_{0}^{1}{[}(-{{t}^{2}}+3t+2)\cdot (-2t)+(2-2{{t}^{2}})\cdot 3+1\cdot 0]dt \\ & \int\limits_{0}^{1}{[}(2{{t}^{3}}-6{{t}^{2}}-4t)+(6-6{{t}^{2}})+0]dt \\ & \int\limits_{0}^{1}{[}2{{t}^{3}}-12{{t}^{2}}-4t+6]dt \\ & =\left[ \frac{2{{t}^{4}}}{4}-\frac{12{{t}^{3}}}{3}-\frac{4{{t}^{2}}}{2}+6t \right]_{0}^{1}=\left[ \frac{2\cdot {{1}^{4}}}{4}-\frac{12\cdot {{1}^{3}}}{3}-\frac{4\cdot {{1}^{2}}}{2}+6\cdot 1 \right]-\left[ \frac{2\cdot {{0}^{4}}}{4}-\frac{12\cdot {{0}^{3}}}{3}-\frac{4\cdot {{0}^{2}}}{2}+6\cdot 0 \right]\left[ \frac{2}{4}-\frac{12}{3}-\frac{4}{2}+6 \right]-0 \\ & \left[ \frac{6}{12}-\frac{48}{12}-\frac{24}{12}+\frac{72}{12} \right]-0=\frac{6-48-24+72}{12}=\frac{6}{12} \\ \end{align}\] \)